4.2.5 Variability of CUR size distributions

Up to now, we have only explored the average CUR size distributions of the toy model and nothing has been said about their variability for different realizations. Although this variability should vanish for infinitely long sequences (CUR size distributions are self-averaging in the thermodynamic limit) there are large fluctuations for finite length molecules. This variability can be determined from the same simulations of the toy model that have been used in the previous sections. Now, for all the realizations we do not only calculate the average CUR size distribution but the standard deviation of the distribution at each value of $ n$. So if we have a collection of distributions obtained from $ n_r$ realizations, $ \{P_i(n)\}$ $ i$=1,$ \dots $,$ n_r$, we can compute the average distribution $ \langle P(n)\rangle$ and the standard deviation of the distribution $ \sigma_{P(n)}$ according to (see Fig. 4.13),

$\displaystyle \langle P(n) \rangle = \frac{1}{n_r} \sum_{i=1}^{n_r} P_i(n)$      
$\displaystyle \langle P(n)^2 \rangle = \frac{1}{n_r} \sum_{i=1}^{n_r} P_i(n)^2$      
$\displaystyle \sigma_{P(n)} = \sqrt{\langle P(n)^2 \rangle - \langle P(n) \rangle^2}~.$     (4.11)

Figure 4.13: CUR size distributions calculated with the toy model. Upper (lower) panel shows the results for a 2252 (6838 bp) sequence. The black curve shows a CUR size distribution averaged over $ n_r=10^4$ realizations. The green region represents the upper and lower limits of the error bars that correspond to the standard deviation of those realizations. Red, blue and orange curves show 3 different realizations. Note the large deviations from the average histogram due to the finite length of the sequences. Simulations were performed with the following parameters: $ k=60$ pN$ \cdot \mu $m$ ^{-1}$, $ \mu =-1.6$ kcal$ \cdot $mol$ ^{-1}$ and $ \sigma =3.2$ kcal$ \cdot $mol$ ^{-1}$.

JM Huguet 2014-02-12